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## geometrical optics

### (7 points)4. Series 37. Year - 4. a perfect passage?

A polarized beam of light coming from a material with refractive index $n_1$ is incident on a planar interface of a material with refractive index $n_2$ such that it does not lose intensity after passing through. It then reaches the parallel interface with refractive index $n_3$, again passing through without any loss, and so on. Find a sequence $n_i$ that satisfies this.

Marek J. met the Brewster's angle

### (12 points)2. Series 37. Year - E. light at the end of a tunel

Measure the illumination intensity of light passing through a cola as a function of the drink's thickness. Determine the absorption coefficient by curve fitting the measured data.

A wasp flew into Jarda's soda can.

### (13 points)6. Series 36. Year - E. ripples

Build an apparatus that can measure the smallest possible ripples on the surface of the liquid. You can choose the container yourself – it can be a cup, a bottle, or something else. Thoroughly describe and take a picture of the whole apparatus. Determine the minimum amplitude you are able to measure.

Karel was staring into space… he was writing his dissertation thesis.

### (7 points)4. Series 36. Year - 4. shot telescope

We have an astronomical (Keplerian) telescope that we want to launch into space. First, however, we will try it on Earth, where we will measure the magnification $Z$. How does the distance between the lenses have to change for it to have the same magnification in space? Lenses have a refractive index of $n$.

Karel gets caught up in those astro-thoughts now and then.

### (10 points)6. Series 35. Year - S. lasering

- How big must an aperture in a spatial filter be if we created it from a lens with a diameter of $40 \mathrm{cm}$ and its focal length is $4 \mathrm{m}$? Our Gaussian laser beam has an input diameter $30 \mathrm{cm}$ and a wavelength $1~053 nm$. The radius of the focus (parameter $\sigma $) of the Gaussian beam can be obtained using

\[\begin{equation*}
r = \frac {2}{\pi }\lambda \frac {f}{D}
\end {equation*}\]
where $D$ is the diameter of the beam, $f$ is the focal length of the lens and $\lambda $ is the wavelength of the laser.

- The laser beam is focused on a surface of a nuclear fuel pellet of a $1 \mathrm{mm}$ diameter. What energy should it have in order for the intensity in its focus to reach $10^{14} W.cm^{-2}$? The radius of the focus is $25 \mathrm{\micro m}$ and a pulse lasts $10 \mathrm{ns}$. How many beams do we need to equally cover the surface of a pellet? What is their total energy?

What energy must the laser beam have if it is not focused on a surface of a nuclear fuel pellet, but the beam diameter matches exactly the diameter of the pellet and the density is its focus reaches $10^{14} W.cm^{-2}$? Assume that we have one such beam and it shines homogenously on the pellet „from all directions“.

### (8 points)3. Series 35. Year - 4. gentle tide

Close to the shore, the speed of sea waves is influenced by the presence of the sea bed. Assume that the speed of waves $v$ is a function of the gravity of Earth $g$ and the water depth $h$. We have $v = C g^\alpha h^\beta $. Using dimensional analysis, determine the speed of the waves as a function of the depth. Constant $C$ is dimensionless, and cannot be determined using this method.

Besides the speed of the waves, swimming Jindra is also interested in the direction of incidence of the waves. Let's define a system of coordinates, where the water surface lies in the $xy$ plane. The shoreline follows the equation $y = 0$, the ocean lies in the $y > 0$ half-plane. The water depth $h$ is given as a function of distance from the shore $h = \gamma y$, where $\gamma = \const $. On the open ocean, where the speed of the waves is constant $c$ (not influenced by the depth), plane waves are propagating at incidence angle $\theta _0$ to the $x$ axis. Find a differential equation \[\begin{equation*} \der {y}{x} = \f {f}{y} \end {equation*}\] describing the shape of the wavefront close to the shore, but do not attempt to solve it, it is far from trivial. Calculate the incidence angle of the wavefront at the shoreline.

**Bonus:** Solve the differential equation and find the shape of the wavefront close to the shore.

Jindra loves simple dimensional analysis and complicated differential equations.

### (3 points)2. Series 35. Year - 1. chasing the light

Jindra walks down a long, lit corridor. His eyes are at a height of $1,7 \mathrm{m}$ above the floor, the light on the ceiling is at a height of $3,4 \mathrm{m}$. Jindra is now at a distance of $10 \mathrm{m}$ (horizontally) from the nearest light and is approaching it at a speed of $3 \mathrm{km\cdot h^{-1}}$. He sees a reflection of the light on the polished floor. How fast is the reflection approaching Jindra at this point?

Jindra remembered walking down the corridor at the elementary school.

### (8 points)2. Series 35. Year - 5. Shkadov thruster

A long time ago in a galaxy far, far away, one civilisation decided to move its whole solar system. One of the possibilities was to build a „Dyson half-sphere“, i. e. a megastructure which would capture approximately half of the radiation output of the start and reflect it in a single direction. An ideal shape would therefore be a paraboloid of revolution. What would be the relation between the radiation output of the star, surface mass density of such a mirror and its distance from the star such that this distance is constant?

Karel watches Kurzgesagt.

### (3 points)2. Series 34. Year - 2. land ahoy

Cathy and Catherine are watching a ship which is sailing with a constant speed towards a port. Cathy is standing on a rock above the port and her eyes are $h_1=20 \mathrm{m}$ above the surface of the water. Catherine is standing under the rock and her eyes are $h_2=1{,}7 \mathrm{m}$ above the surface of the water. If Catherine sees the top of the incoming ship $t=25 \mathrm{min}$ after Cathy sees it, what is the time of arrival of the ship to the port? Assume that the Earth is a perfect sphere with a radius $r=6378 \mathrm{km}$.

Radka remembered a vacation by the sea.